Absolutely Continuous Measures: Difference between revisions
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==Definitions== | ==Definitions== | ||
Let <math> (X,\mathcal{M},\mu_1) </math> be a measure space. The measure <math> \mu_2:\mathcal{M}\rightarrow [0,\infty] </math> is said to be absolutely continuous with respect to the measure <math> \mu_1 </math> if we have that <math> | Let <math> (X,\mathcal{M},\mu_1) </math> be a measure space. The measure <math> \mu_2:\mathcal{M}\rightarrow [0,\infty] </math> is said to be absolutely continuous with respect to the measure <math> \mu_1 </math> if we have that <math> \mu_2(T) = 0 </math> for <math> T\in \mathcal{M}</math> such that <math> \mu_1(T) = 0</math> (see [1]). | ||
<in progress> | <in progress> | ||
==References== | ==References== | ||
[1]: Taylor, M.E. "Measure Theory and Integration". 50-51. | |||
Revision as of 17:49, 18 December 2020
Definitions
Let be a measure space. The measure is said to be absolutely continuous with respect to the measure if we have that for such that (see [1]).
![{\displaystyle \mu _{2}:{\mathcal {M}}\rightarrow [0,\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98db72c9313a5f1a8af79396881782bea9858b9d)
<in progress>
References
[1]: Taylor, M.E. "Measure Theory and Integration". 50-51.